Calculate the rent of an increasing annuity at 4% interest compounded semiannually if payments are made every half-year and the future value after 7 years is $10,000

Calculate the rent of an increasing annuity at 4% interest compounded semiannually if payments are made every half-year and the future value after 7 years is $10,000.

The correct answer and explanation is:

To calculate the rent (payment amount) of an increasing annuity, we use the formula for the future value (FV) of an increasing annuity:

📌 Formula:

FV=R[(1+i)n−1i−n(1+i)n]FV = R \left[ \frac{(1 + i)^n – 1}{i} – \frac{n}{(1 + i)^n} \right]

Where:

• FVFV = future value of the annuity
• RR = rent (the amount of the first payment; subsequent payments increase by R)
• ii = interest rate per period
• nn = total number of payments

✅ Step 1: Identify values

• Future Value FV=10,000FV = 10{,}000
• Interest rate =4%= 4\% compounded semiannually → i=0.04/2=0.02i = 0.04 / 2 = 0.02
• Number of years = 7, with semiannual payments → n=7×2=14n = 7 \times 2 = 14

✅ Step 2: Plug into formula

10,000=R[(1+0.02)14−10.02−14(1+0.02)14]10{,}000 = R \left[ \frac{(1 + 0.02)^{14} – 1}{0.02} – \frac{14}{(1 + 0.02)^{14}} \right]

Calculate:

• (1.02)14≈1.319(1.02)^{14} \approx 1.319
• (1.319−1)0.02=0.3190.02=15.95\frac{(1.319 – 1)}{0.02} = \frac{0.319}{0.02} = 15.95
• 141.319≈10.61\frac{14}{1.319} \approx 10.61

So: 10,000=R(15.95−10.61)=R(5.34)10{,}000 = R \left(15.95 – 10.61\right) = R (5.34)

✅ Step 3: Solve for R

R=10,0005.34≈1,873.60R = \frac{10{,}000}{5.34} \approx 1{,}873.60

✅ Final Answer:

R=1,873.60\boxed{R = 1{,}873.60}

🧠 Explanation (300 words):

An increasing annuity is a type of financial arrangement where payments are made periodically, and each payment is larger than the previous by a fixed amount — often equal to the initial rent. This is commonly used in structured savings or retirement plans, where contributions grow over time.

In this problem, you are told that $10,000 is the future value (FV) of an increasing annuity after 7 years. Since payments are made every 6 months, the compounding period is semiannual, and the interest rate is 4% annually, or 2% per half-year.

Over 7 years, with semiannual contributions, there will be 14 payments in total. Because each payment is increasing, and not level, we cannot use the formula for a regular annuity. Instead, we use the future value formula specifically designed for an increasing annuity.

This formula accounts for the fact that earlier payments have more time to accumulate interest than later ones, and each payment is larger than the last. The difference between this and the standard annuity formula is the subtraction of a discounting term to reflect the increase in payments.

After plugging in all known values and solving the equation, we find that the first payment (R) — or the rent — must be approximately $1,873.60 in order for the future value of the increasing annuity to total $10,000 after 7 years at a 4% annual interest rate compounded semiannually.